Suehiro and his colleagues have devised the Suehiro's DFT (OSDM) system that is a new information transmitting system using the Kronecker product of the row vectors (The “column vectors” may also be used instead of the “row vectors”. The row vectors are used in the description of this specification) of a DFT (Discrete Fourier Transform) matrix and the data vectors (see Non-Patent Documents 1 and 2).
It is recognized that the wireless frequency usage efficiency of this system is about two times higher than that of the OFDM (Orthogonal Frequency Division Multiplex) system that is used today in various types of communication (see Non-Patent Document 3).
Next, the following describes the OSDM (Orthogonal Signal Division Multiplex) system that is a transmitting method for transmitting/receiving signals having the length M×N generated by calculating the Kronecker product of each of the N row vectors (N is a natural number equal to or larger than 3) of an Nth order DFT matrix and data having the length M (M is a natural number equal to or larger than 2).
(DFT Matrix and Transmission Signal)
First, the following describes an Nth order DFT (Discrete Fourier Transform) matrix.
Let the Nth order DFT matrix FN be defined as follows.FN=[fN(i,j)]  (1)
where the Nth order inverse DFT matrix FN−1 is the complex conjugate of the DFT matrix FN.
In the above expression, i is a row number (0≦i≦N−1) and j is a column number (0≦j≦N−1).fN(i,j)=exp(2π√{square root over ( )}−1ij/N)/√{square root over ( )}N  (2)
The variable WN corresponding to the point, generated by dividing a unit circle into N, is defined as follows as shown in FIG. 1.WN≡exp(2π√{square root over ( )}−1)/N  (3)
When this WN is used, the DFT matrix FN is as shown in FIG. 2.
WN is a rotor and the following relation is satisfied.WNN=ej2π=1  (4)WNN−k=WN2N−k= . . . =WN−k  (5)As shown in FIG. 2, the Nth order DFT matrix FN has N row vectors, that is, vector fN,0, vector fN,1 . . . vector fN,N−1. The cyclic crosscorrelations among those row vectors are zero in all shifts.
Next, the following describes data transmission using those row vectors. As shown in FIG. 3, the signals S0, S1 . . . SN−1 are generated from N pieces of transmission data each having the length M (data X0(x00, x01, . . . , x0(M−1)), data X1 (x10, x11, . . . , x1(M−1)), . . . , data X(N−1) (x(N−1)0, x(N−1)1, . . . , x(N−1)(M−1))), received from transmitting unit #0, transmitting unit #1 . . . transmitting unit #(N−1) using row vector fN,0, row vector fN,1, . . . , row vector fN,N−1, and the generated signals are transmitted.
                                                                        S                0                            =                                                f                                      N                    ,                    0                                                  ⊗                                  DataX                  0                                                                                                                        S                1                            =                                                f                                      N                    ,                    1                                                  ⊗                                  DataX                  1                                                                                          ⋮                                                                              S                                  N                  -                  1                                            =                                                f                                      N                    ,                                          N                      -                      1                                                                      ⊗                                  DataX                                      (                                          N                      -                      1                                        )                                                                                                          [                  Mathematical          ⁢                                          ⁢          expression          ⁢                                          ⁢          1                ]            
where  is the Kronecker product. Transmitting the generated signals S0, S1 . . . SN−1 allows data to be transmitted from multiple transmitting units without correlation. Note that the length of transmitted signals is N×M.
That is, because the cyclic crosscorrelation between any two signals of the signals S0, S1 . . . SN−1 is zero in all shifts, the well-designed matched filters allows data sequences to be separated at reception time even when the signals are added up.
(Matched Filter)
The vector IM(1, 0, . . . , 0) having the length M is defined.
Here, the matched filters for matching to the signals of the Kronecker product of the vectors fk(0≦k≦N−1) and IM are provided.[Mathematical expression 2]fkIM=(WN0,0, . . . ,0,WNk,0, . . . ,0,WN(N−1)k,0, . . . 0)/√{square root over ( )}N  (7)
When the signals Sk(0≦k≦N−1) are input to the matched filters, M units of data in the center of the output becomes data XK.
In addition, when the signals Sg (where g≠k, 0≦k≦N−1, 0≦g≦N−1) are input to the matched filters[Mathematical expression 3]fkIM=(WN0,0, . . . ,0,WNk,0, . . . ,0,WN(N−1)k,0, . . . 0)/√{square root over ( )}N  (8)
The M units of data in the center of the output signal are always 0. This means that, even when the signals, from signal S0 to signal SN−1, are added up, only XK is produced when they are input to the matched filters of fKIM.
(Pseudo-Periodic Signal)
Let Ssum be the signal produced by adding up signal S0 to signal SN−1. Because the signal Ssum is a limited-length sequence having the length MN, the periodicity obtained by the DFT matrix is lost when the signal is input to the multipath channels. In such a case, the data Xk(0≦k≦N−1) cannot be obtained from the matched filter output.
Multipath channels do not affect the periodicity of the signal if the signal is a periodic signal having an unlimited length. However, transmitting a sequence having an unlimited length is not practical. To solve this problem, a pseudo-periodic signal, generated by selecting a signal having a necessary length from the periodic sequence of an unlimited length, is used.
First, let L2 be a value larger than the assumed multipath delay time.
When there is no direct-path signal or when the power level of the direct-path signal is extremely low, the delay time for the maximum amplitude signal becomes sometimes negative. Let L1 be a value considering that time.
Using those values L1 and L2, the pseudo-periodic signal, such as the one shown in FIG. 4, is generated and transmitted.
The part corresponding to L2 is called a cyclic prefix, and the part corresponding to L1 is called a cyclic postfix. At reception time, both prefixes must be removed before the signal enters the matched filter.
(Pilot Signal)
The data sequence X0 is defined as follows where the length is M.X0=(1,0,0,0, . . . ,0)  (9)[Mathematical expression 4]
When “fkIM” is calculated using this data sequence and the data sequence is input directly into fkIM, the central part of the output becomes as follows.X0=(1,0,0,0, . . . ,0)  (10)
Next, S0 is converted to a pseudo-periodic signal, which is sent via multipath channels. When the cyclic (pre/post) prefixes are removed and the signal is input to the matched filters of fkIM, the M units of data in the central part of the output are as follows.X0=(p0,p1,p2,p3, . . . ,p(L2−1),0,0, . . . ,0)  (11)
where (p0, p1, p2, p3, . . . , pk, . . . , p(L2−1)) are complex coefficients that are multiplied by the paths which arrived with a delay of time k. They correspond to the transmission characteristics including the transmission characteristics of the transmitting device, the transmission characteristics of the propagation space, and the transmission characteristics of the receiving device and represent the channel characteristics on the time axis.
This pk is usually represented as shown below using the amplitude coefficient rk and the phase rotation θk.pk=rk·ejθk  (12)
As the pilot signal, the signal of the ZACZ (Zero Auto Correlation Zone Sequence) sequence, the signal of the ZCCZ (Zero Crosscorrelation Zone Sequence) sequence, and the signal of the PN sequence may be used.[Mathematical expression 5]
In this case, the output of the matched filters of fkIM described above must be input to the matched filters that match to those pilot signals.
Also when ZACZ and so on are used as the pilot signal, the channel characteristics on the time axis, including the multipath characteristics, may be detected.
(Simultaneous Equation)
As described above, the channel characteristics on the time axis, including the multipath characteristics, can be obtained by inserting the pilot signal.
The M units of data (dk0-dk(M−1)) in the center of each matched filter output of the data signal parts Xk(1<k<N−1), other than the pilot, have the relation between the data and the multipath characteristics which is shown by the following expression.
                                                                                          (                                                            p                      0                                        ,                                          p                      1                                        ,                    …                    ⁢                                                                                  ,                                          p                                                                        L                          ⁢                                                                                                          ⁢                          2                                                -                        2                                                              ,                                          p                                                                        L                          ⁢                                                                                                          ⁢                          2                                                -                        1                                                              ,                    0                    ,                    …                    ⁢                                                                                  ,                    0                    ,                    0                    ,                    0                                    )                                ·                                  x                                      k                    ⁢                                                                                  ⁢                    0                                                              +                                                                                                            (                                      0                    ,                                          p                      0                                        ,                                          p                      1                                        ,                    …                    ⁢                                                                                  ,                                          p                                                                        L                          ⁢                                                                                                          ⁢                          2                                                -                        2                                                              ,                                          p                                                                                                    L                            ⁢                                                                                                                  ⁢                            2                                                    -                          1                                                ⁢                                                                                                                                        ,                    0                    ,                    …                    ⁢                                                                                  ,                    0                    ,                    0                                    )                                ·                                  x                                      k                    ⁢                                                                                  ⁢                    1                                                              +                                                                                                            (                                      0                    ,                    0                    ,                                          p                      0                                        ,                                          p                      1                                        ,                    …                    ⁢                                                                                  ,                                          p                                                                        L                          ⁢                                                                                                          ⁢                          2                                                -                        2                                                              ,                                          p                                                                        L                          ⁢                                                                                                          ⁢                          2                                                -                        1                                                              ,                    0                    ,                    …                    ⁢                                                                                  ,                    0                                    )                                ·                                  x                                      k                    ⁢                                                                                  ⁢                    2                                                              +                                                            ⋮                                                                                                +                                      (                                          0                      ,                      0                      ,                      0                      ,                      …                      ⁢                                                                                          ,                      0                      ,                      0                      ,                      0                      ,                      0                      ,                                              p                        0                                            ,                                              p                        1                                                              )                                                  ·                                  x                                      k                    ⁡                                          (                                              M                        -                        2                                            )                                                                                  +                                                                                                            (                                      0                    ,                    0                    ,                    0                    ,                    0                    ,                    …                    ⁢                                                                                  ,                    0                    ,                    0                    ,                    0                    ,                    0                    ,                                          p                      0                                                        )                                ·                                  x                                      k                    ⁡                                          (                                              M                        -                        1                                            )                                                                                  =                                                                          (                                                d                                      k                    ⁢                                                                                  ⁢                    0                                                  ,                                  d                                      k                    ⁢                                                                                  ⁢                    1                                                  ,                                  d                                      k                    ⁢                                                                                  ⁢                    2                                                  ,                …                ⁢                                                                  ,                                  d                                      k                    ⁡                                          (                                              M                        -                        2                                            )                                                                      ,                                  d                                      k                    ⁡                                          (                                              M                        -                        1                                            )                                                                                  )                                                          (        13        )            This is expressed by the matrix shown in Expression (14) given below.
                                              ⁢                  [                      Mathematical            ⁢                                                  ⁢            expression            ⁢                                                  ⁢            6                    ]                                                                              [                                                                      d                                      k                    ⁢                                                                                  ⁢                    0                                                                                                                        d                                      k                    ⁢                                                                                  ⁢                    1                                                                                                                        d                                      k                    ⁢                                                                                  ⁢                    2                                                                                                      ⋮                                                                                      d                                      k                    ⁡                                          (                                                                        L                          2                                                -                        1                                            )                                                                                                                                            d                                      kL                    2                                                                                                      ⋮                                                                                      d                                      k                    ⁡                                          (                                              M                        -                        1                                            )                                                                                                    ]                =                                                       [                                                                                          P                      0                                                                            0                                                        0                                                        …                                                        0                                                        0                                                        …                                                                              P                      1                                                                                                                                  P                      1                                                                                                  P                      0                                                                            0                                                        …                                                        0                                                        0                                                        …                                                                              P                      2                                                                                                                                  P                      2                                                                                                  P                      1                                                                                                  P                      0                                                                            …                                                        0                                                        0                                                        …                                                                              P                      3                                                                                                                                                                                                              ⋮                                                                                                                                                          ⋮                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      P                                                                        L                          2                                                -                        1                                                                                                                        P                                                                        L                          2                                                -                        2                                                                                                                        P                                                                        L                          2                                                -                        3                                                                                                  …                                                                              P                      0                                                                            0                                                        …                                                        0                                                                                        0                                                                              P                                                                        L                          2                                                -                        1                                                                                                                        P                                                                        L                          2                                                -                        2                                                                                                  …                                                                              P                      1                                                                                                  P                      0                                                                            …                                                        0                                                                                                                                                                                          ⋮                                                                                                                                                          ⋮                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                0                                                        0                                                        0                                                        …                                                        0                                                        0                                                        …                                                                              P                      0                                                                                  ]                        ⁡                          [                                                                                          I                                              k                        ⁢                                                                                                  ⁢                        0                                                                                                                                                        I                                              k                        ⁢                                                                                                  ⁢                        1                                                                                                                                                        I                                              k                        ⁢                                                                                                  ⁢                        2                                                                                                                                  ⋮                                                                                                              I                                              k                        ⁡                                                  (                                                                                    L                              2                                                        -                            1                                                    )                                                                                                                                                                                I                                              kL                        2                                                                                                                                  ⋮                                                                                                              I                                              k                        ⁡                                                  (                                                      M                            -                            1                                                    )                                                                                                                                ]                                                          (        14        )            where
                              [                      Mathematical            ⁢                                                  ⁢            expression            ⁢                                                  ⁢            7                    ]                ⁢                                  ⁢        If                                                            P        =                  [                                                                      P                  0                                                            0                                            0                                            …                                            0                                            0                                            …                                                              P                  1                                                                                                      P                  1                                                                              P                  0                                                            0                                            …                                            0                                            0                                            …                                                              P                  2                                                                                                      P                  2                                                                              P                  1                                                                              P                  0                                                            …                                            0                                            0                                            …                                                              P                  3                                                                                                                                                                  ⋮                                                                                                                          ⋮                                                                                                                                                                                                                          ⋮                  ⁢                                                                                                                                                                                                                                            P                                                            L                      2                                        -                    1                                                                                                P                                                            L                      2                                        -                    2                                                                                                P                                                            L                      2                                        -                    3                                                                              …                                                              P                  0                                                            0                                            …                                            0                                                                    0                                                              P                                                            L                      2                                        -                    1                                                                                                P                                                            L                      2                                        -                    2                                                                              …                                                              P                  1                                                                              P                  0                                                            …                                            0                                                                                                                                                  ⋮                                                                                                                          ⋮                                                                                                                                                                                                                                                                                  ⁢                  ⋮                                                                                                                                                                  0                                            0                                            0                                            …                                            0                                            0                                            …                                                              P                  0                                                              ]                                    (        15        )                                          D          k                =                  [                                                                      d                                      k                    ⁢                                                                                  ⁢                    0                                                                                                                        d                                      k                    ⁢                                                                                  ⁢                    1                                                                                                                        d                                      k                    ⁢                                                                                  ⁢                    2                                                                                                      ⋮                                                                                      d                                      k                    ⁡                                          (                                                                        L                          2                                                -                        1                                            )                                                                                                                                            d                                      kL                    2                                                                                                      ⋮                                                                                      d                                      k                    ⁡                                          (                                              M                        -                        1                                            )                                                                                                    ]                                    (        16        )            then,Dk=PtXk  (17)
Solving Expression (17) for Xk allows the receiving side to obtain the transmission data for which the channel characteristics on the time axis are compensated, wherein the channel characteristics include the transmission characteristics of the transmitting device side, the transmission characteristics of the propagation space, and the transmission characteristics of the receiving device side.
To solve this simultaneous equation simply, the both sides of Expression (17) are multiplied by the inverse matrix of P from the left.
                                          P                          -              1                                ⁢                      D            k                          =                                            P                              -                1                                      ⁢                          P              t                        ⁢                          X              k                                =                      X            k                                                    t                                              (        18        )                Non-Patent Document 1: N. Suehiro, C. Han, T. Imoto, and N. Kuroyanagi, “An information transmission method using Kronecker product”, Proceedings of the IASTED International Conference Communication Systems and Networks, pp. 206-209, September 2002.    Non-Patent Document 2: N. Suehiro, C. Han, and T. Imoto, “Very Efficient wireless usage based on pseudo-coherent addition of multipath signals using Kronecker product with rows of DFT matrix”, Proceedings of International Symposium on Information Theory, pp. 385, June 2003.    Non-Patent Document 3: Naoki Suehiro, Rongzhen Jin, Chenggao Han, Takeshi Hashimoto, “Performance of Very Efficient Wireless Frequency Usage System Using Kronecker Product with Rows of DFT Matrix”, Proceedings of 2006 IEEE Information Theory Workshop (ITW '06), pp. 526-529, October 2006.